• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

History of statistics wikipedia, lookup

Regression toward the mean wikipedia, lookup

Student's t-test wikipedia, lookup

Taylor's law wikipedia, lookup

Bootstrapping (statistics) wikipedia, lookup

Resampling (statistics) wikipedia, lookup

Misuse of statistics wikipedia, lookup

Psychometrics wikipedia, lookup

Foundations of statistics wikipedia, lookup

Fisher–Yates shuffle wikipedia, lookup

Transcript
```• For a permutation test, we have H0:
F1(x) = F2(x) vs
Ha: F1(x) <= F2(x)
• Note this alternative means that the
density of the first population is
larger than that of the second...
sketch to see this!
• Special case is the shift alternative:
Ha: F1(x) = F2(x-D), where D > 0.
Sketch this!
• We may also have the alternative Ha:
F1(x) >= F2(x) and the two-sided
alternative
Ha: F1(x) <= F2(x) or F1(x) >= F2(x),
for all x, with strict inequality
occurring for at least one x. For the
shift alternative, this is D ne 0.
Think of D as the difference between
medians of the two populations...
• Permutation tests may also be
performed on other statistics besides
the mean – of course, if population(s)
are normal then mean are probably
best – the textbook mentions ones
based on the median and the trimmed
mean. This gives permutation tests
much flexibility ...
• Note in Table 2.2.1 that changing the
max. value has no effect on the
medians but could impact the mean
• p-values obtained from permutation
distributions of test statistics are
exact in the sense that they are not
dependent upon unverified
population distribution ...
• Approximate p-values may be
obtained from random sampling of
permutations and for large number of
random samples, the error can be
quite small – see bottom of page 32
for margin of error...
Permutation tests
• We may also get approximate p-values by randomly
sampling the permutations, instead of trying to write
them all down. This is useful when m+n is large…
• Do as before:
– assign experimental units to the two groups at
random and compute the difference between the
means of the two groups, Dobs . There are m units
assigned to group1 and n units to group2 (m+n total
units).
– randomly "sample" all the m+n observations so there
are m in group1 and n in group2.
– compute the difference between the means of the two
groups of the "sampled" vector, D.
– repeat this procedure a large number of times (1000
or larger). For an upper-tailed test, calculate the
empirical p-value: # of times D>= Dobs / 1000